Equation in positive integer

Nené · Poincare Conjecture Offline Joined: 19 Jan 2005 Posts: 170

Find solution in positive integer of equation:

$x^2 + y^2 = z^{2l + 1} + z$

wth l is also a positive integer

**Posted:** Sun Jan 30, 2005 12:00 pm

blang · Poincare Conjecture Offline Joined: 13 Jan 2005 Posts: 157

$z(z^{2l}+1)$ is a sum of two squares iff $z$ is a sum of two squares.
We can take $(x,y,z)=(t(t^2+u^2)^l-u,t+u(t^2+u^2)^l,t^2+u^2)$ ( $u,t$ integers), wich give an example...

**Posted:** Sun Jan 30, 2005 12:56 pm

_________________
blang (sorry for my french english!)

Myth

I see we have an expert in Number Theory now Wink

**Posted:** Sun Jan 30, 2005 12:58 pm

_________________
Myth is out of here

filip

**Posted:** Sun Jan 30, 2005 1:01 pm

grobber

A number is the sum of two squares iff every prime divisor of the form $4k+3$ appears with an even exponent. This is true for $z^{2l}+1$ , which is a sum of two squares, so it's true for $z(z^{2l}+1)$ iff it's true for $z$ .

**Posted:** Sun Jan 30, 2005 1:04 pm

Myth

You are so fast, grobber Sad

I was writing reply with the same phrase too Wink

**Posted:** Sun Jan 30, 2005 1:06 pm

_________________
Myth is out of here

kueh · Riemann Hypothesis Offline Joined: 26 May 2004 Posts: 417

**Posted:** Sun Jan 30, 2005 2:43 pm

grobber

I think it has been discussed quite a few times on ML. The necessity follows from the fact that if a prime of the form $4k+3$ divides $a^2+b^2$ , then it divides both $a$ and $b$ . The sufficiency follows from the fact that each prime of the form $4k+1$ can be written as the sum of to squares and the product of sums of two squares is also a sum of two squares.

Sorry I don't give the proofs, but, as I said, it's been discussed before. You could also search for "sum of two squares" on Google, for example. You'll find lots of stuff I could not explain properly Smile

.

**Posted:** Sun Jan 30, 2005 2:52 pm

pbornsztein

**Posted:** Sun Jan 30, 2005 3:33 pm

blang · Poincare Conjecture Offline Joined: 13 Jan 2005 Posts: 157

I would like to be an expert in number theory. Mr. Green

**Posted:** Sun Jan 30, 2005 10:41 pm

_________________
blang (sorry for my french english!)